Probability, Percent, Rational Number Equivalence Math A College Prep
Module #1 Student Edition 2017-2018
Created in collaboration with Utah Middle School Math Project A University of Utah Partnership Project
San Dieguito Union High School District
Table of Contents - Module #1: Probability, Percent, Rational Number Equivalence MODULE 1: PROBABILITY, PERCENT, RATIONAL NUMBER EQUIVALENCE ........................................................... 3 STANDARDS FOR MATHEMATICAL PRACTICE: A GUIDE FOR STUDENTS AND PARENTS ...................................... 5 SECTION 1.1: INVESTIGATE CHANCE PROCESSES. DEVELOP/USE PROBABILITY MODELS.* ................................... 6 1.0A LESSON: REVIEW FROM EARLIER GRADES*............................................................................................ 7 1.0B LESSON: “10 X 10 GRIDS” & CONVERSION* ............................................................................................ 8 1.0C LESSON: CONVERTING BETWEEN FRACTIONS, DECIMALS, AND PERCENTS* .......................................... 11 1.1A LESSON: PROBABILITY PREDICTIONS* .................................................................................................. 13 1.1B LESSON: PROBABILITY – RACE TO THE TOP* ........................................................................................ 18 1.1C LESSON: SAMPLE SPACES, OUTCOMES, AND PROBABILITIES* ............................................................... 20 SECTION 1.2: UNDERSTAND/APPLY EQUIVALENCE IN RATIONAL NUMBER FORMS. CONVERT BETWEEN FORMS (FRACTION, DECIMAL, PERCENT).* ................................................................................................................... 23 1.2A LESSON: BAR MODELS WITH FRACTIONS AND DECIMALS* ..................................................................... 24 1.2B LESSON: RATIONAL NUMBER ORDERING AND ESTIMATION* .................................................................. 26 1.2C LESSON: PROBABILITY, FRACTIONS, PERCENTAGE, & RATIO* ............................................................... 29 1.2D LESSON: RATIONAL NUMBERS IN APPLICATIONS WITH MODELS* ........................................................... 32 SECTION 1.3: SOLVE PERCENT PROBLEMS INCLUDING DISCOUNTS, INTEREST, TAXES, TIPS, AND PERCENT INCREASE OR DECREASE.* .............................................................................................................................. 35 1.3A LESSON MODEL PERCENT AND FRACTION PROBLEMS* ......................................................................... 36 1.3B LESSON: PERCENT AND FRACTION PROBLEMS* .................................................................................... 39 * Denotes a lesson that was adapted from Utah Middle School Math Project. © Utah Middle School Math Project and University of Utah Partnership http://utahmiddleschoolmath.org/ This work is licensed under a Creative Commons Attribution-NonCommercial 2.5 Generic License http://creativecommons.org/licenses/by-nc/2.5/ This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/legalcode
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Module 1: Probability, Percent, Rational Number Equivalence Online support for this module can be found at http://goo.gl/s2vDtM (case sensitive) or using the QR code below. This website includes copies of student classwork, homework, and instructional videos for common core standards.
Common Core Standard(s) Number Sense: Curriculum Support Website 1. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. 7.NS.2d 2. Solve real-world and mathematical problems involving the four operations with rational numbers. 7.NS.3 Probability and Statistics: 1. Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ½ indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. 7.SP.5 2. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. 7.SP.6 Equations and Expressions: 1. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. 7.EE.3 Module 1 Summary: Module 1 begins with a brief introduction to probability as a means of reviewing and applying arithmetic with whole numbers and fractions. In addition to covering basic counting techniques and listing outcomes in a sample space, students distinguish theoretical probabilities from experimental approaches to estimate probabilities. Another reason for starting the year with probability activities is to develop a culture of investigation, discussion and collaboration in the classroom. Throughout the module students are provided with opportunities to review and build fluency with fractions, percents, and decimals from previous grades. Students should understand that fractions, percent and decimals are all relative to a whole. Students will also compare and order fractions (both positive and negative.) This module concludes with a section specifically about solving percent and fraction problems, including those involving discounts, interest, taxes, tips, and percent increase or decrease. Vocabulary: chance, decimal, fraction, frequency, experimental probability, percent, probability, ratio, theoretical probability
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Connections to Content: Prior Knowledge In previous coursework, students developed the concept of a ratio. Thus students should be familiar with the idea of part:whole, part:part, and whole:whole relationships. It is important to emphasize that in this module only part:whole relationships are discussed (probability is a part:whole relationship as are fractions, decimals and percents.) Later in 7th grade students will discuss “odds” which are part:part relationships. Students have used all four operations (addition, subtraction, multiplication, and division) when working with fractions and decimals in prior grades. They should have used both number line and bar models to represent fractions, percent and decimals. In 6th grade students placed both positive and negative numbers on a number line, however they do not operate on negative numbers until 7th grade (this will take place in Module 2). Future Knowledge As students move through this module, they will begin by studying probability (this module is only an introduction to probability, students will work more with probability in Module 8.) The concepts learned in 7th grade around chance processes and theoretical and experimental probabilities will be extended in later courses when students study conditional probability, compound events, evaluate outcomes of decisions, use probabilities to make fair decisions, etc. While studying probability students will continue their study of rational numbers. They will convert rational numbers to decimals and percents and will look at their placement on the number line. This lays the foundation for 8th grade where students study irrational numbers to complete the Real Number system.
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Standards for Mathematical Practice: A Guide for Students and Parents The Standards for Mathematical Practices are central to the Common Core. These practices build fluency and help students become better decision-makers and problem solvers. The practices reflect the most advanced and innovative thinking on how students should interact with math content. Students and parents will develop skill with these standards by asking some of these questions: Make Sense of Problems and Persevere in Solving Them. • What is the problem that you are solving for? • Can you think of a problem that you recently solved that is similar to this one? • How will you go about solving the problem? (i.e. What’s your plan?) • Are you progressing towards a solution? How do you know? Should you try a different solution plan? • How can you check your solution using a different method? Construct Viable Arguments and Critique the Reasoning of Others. • Can you write or recall an expression or equation to match the problem situation? • What do the numbers or variables in the equation refer to? • What’s the connection among the numbers and variables in the equation? Reason Abstractly and Quantitatively. • Tell me what your answers(s) mean(s) • How do you know that your answer is correct? • If I told you I think the answer should be (a wrong answer), how would you explain to me why I’m wrong? Model with Mathematics. • Do you know a formula or relationship that fits this problem situation? • What’s the connection among the numbers in the problem? • Is your answer reasonable? How do you know? • What do(es) the number(s) in your solution refer to? Use Appropriate Tools Strategically. • What tools could you use to solve this problem? How can each one help you? • Which tool is most useful for this problem? Explain your choice. • Why is this tool (the one selected) better to use than (another tool mentioned)? • Before you solve the problem, can you estimate the solution? Attend to Precision. • What do the symbols that you used mean? • What units of measure are you using (for measurement problems) • Explain to me what (term from the lesson) means. Look For and Make Use of Structure. • What do you notice about the answers to the exercises you’ve just completed? • What do different parts of the expression or equation you are using tell you about possible correct answers? Look for and Express Regularity in Repeated Reasoning. • What shortcut can you think of that will always work for these kinds of problems? • What pattern(s) do you see? Can you make a generalization?
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Section 1.1: Investigate chance processes. Develop/use probability models.* Section Overview: This is students’ first formal introduction to probability. In this section students will study chance processes, which concern experiments or situations where they know which outcomes are possible, but they do not know precisely which outcome will occur at a given time. They will look at probabilities as ratios expressed as fractions, decimals, or percents (part:whole). Probabilities will be determined by considering the results or outcomes of experiments. They will learn that the set of all possible outcomes for an experiment is a sample space. They will recognize that the probability of any single event can be expressed in terms of impossible, unlikely, equally likely, likely, certain, or as a number between 0 and 1, inclusive. Students will focus on two concepts in probability of an event: experimental and theoretical. They will understand the commonalities and differences between experimental and theoretical probability in given situations. Concepts and Skills to be Mastered (from standards) 1. 2. 3. 4.
Understand and apply likelihood of a chance event as between 0 and 1. Approximate probability by collecting data on a chance process (experimental probability). Calculate theoretical probabilities on a chance process. Given the probabilities (different scenarios in a chance process), predict the approximate frequencies for those scenarios (if experimenting on a chance process). 5. Use appropriate fractions, decimals and percents to express the probabilities.
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1.0A Lesson: Review From Earlier Grades* Name:
Period:
Fraction
Decimal
Percent
Bar Model
A.
B.
C.
D.
E.
F.
1. Use a bar model to represent
2. Use a bar model to represent
3. Use a bar model to represent
! !
! !
of a whole.
of a whole.
! !"
of a whole.
4. What do you notice about the fractions in #1-3?
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1.0B Lesson: “10 x 10 Grids” & Conversion* Name:
Period:
Write the equivalent values for the following parts of a candy bar. The bar on the left is divided into two equal parts. 1 part =_____(fraction) =_____(decimal) =_____ (percent) 2 parts =_____(fraction) =_____(decimal) =_____ (percent)
The bar on the left is divided into three equal parts. 1 part =_____(fraction) =_____(decimal) =_____ (percent) 2 parts =_____(fraction) =_____(decimal) =_____ (percent)
The bar on the left is divided into four equal parts. 1 part =_____(fraction) =_____(decimal) =_____ (percent) 2 parts =_____(fraction) =_____(decimal) =_____ (percent) 3 parts =_____(fraction) =_____(decimal) =_____ (percent)
The bar on the left is divided into five equal parts. 1 part =_____(fraction) =_____(decimal) =_____ (percent) 3 parts =_____(fraction) =_____(decimal) =_____ (percent)
The bar on the left is divided into six equal parts. 1 part =_____(fraction) =_____(decimal) =_____ (percent) 2 parts =_____(fraction) =_____(decimal) =_____ (percent) 3 parts =_____(fraction) =_____(decimal) =_____ (percent)
The bar on the left is divided into eight equal parts. 2 parts =_____(fraction) =_____(decimal) =_____ (percent) 7 parts =_____(fraction) =_____(decimal) =_____ (percent)
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1. To the left is a 10 × 10 Grid. Why do you think it is called a 10 × 10 grid?
2. How could you use the 10 × 10 grid to show the fraction
6 ? Explain. 100
3. Shade this fraction in the grid to the left.
4. What fraction is shown in this 10 × 10 grid? Explain.
5. What is the decimal equivalent for this fraction?
6. What fraction is shown in this 10 × 10 grid?
7. What is the decimal equivalent for this fraction?
8. Shade the given decimal in each grid below: a. 0.27
b. 0.4
c. 0.125
9. Write the fraction represented in each model above.
a. __________
b. __________
c. __________
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10. Shade the fractional part of each grid. Then write the fraction as a decimal and a percent.
a.
!
b.
!
!
c.
!"
! !
decimal: __________
decimal: __________
decimal: __________
percent: __________
percent: __________
percent: __________
d.
!
e.
!""
!
f.
!"
! !
decimal: __________
decimal: __________
decimal: __________
percent: __________
percent: __________
percent: __________
11. Use long division to show how you can convert each fraction to a decimal and then a percent. a.
! !"
b.
! !
c.
! !
SDUHSD Math A College Prep Module #1 – STUDENT EDITION 2017-2018
d.
! !
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1.0C Lesson: Converting Between Fractions, Decimals, and Percents* Name:
Period:
1. Fill in the boxes below to show how you can convert between fractions, decimals, and percentages. How can you write a percent as a fraction?
How can you write a fraction as a decimal?
How can you write a decimal as a percent?
Show how to use the steps you described above to complete the following problems. 2. Write 45% as a fraction in simplest form. 5. Write 0.45 as a percent. 3. Write
4. Write
3 as a decimal. 5
6. Write 1 as a percent.
1 as a decimal. 9
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Fill in each blank with the equivalent fraction, decimal or percent. Use bar notation for repeated decimals. Fraction 7. 8.
Decimal
0.42
10.
0.8
11.
32%
9 20
13.
21%
14.
0.06
15. 16.
7%
1 8
17.
0.99
18. 19.
75%
1 4
20. 21.
20%
6 15
22.
1.5
23.
250 %
24. 25. 26.
Show your work over here!
6 10 4 25
9.
12.
Percent
3.0
8 11 2 3
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1.1A Lesson: Probability Predictions* Name:
Period:
1. In your own words, what do you think these terms mean? a. “Experimental” probability: b. “Theoretical” probability: 2. We will examine experimental and theoretical probability in this activity. Your bag contains a total of 12 green and blue tiles. Without looking in the bag, you will be making a guess as to how many GREEN tiles the bag contains. **DO NOT LOOK IN THE BAG UNTIL YOU ARE TOLD TO!** a. Draw a marble/tile (without looking in the bag), record the color, replace it, redraw, record, replace. Do this 6 times. Draw # 1 2 3 4 5 6 Color Based on your experiment, what portion of the 6 draws were GREEN? ________ Based on your experiment of 6 draws, how many green tiles do you think are in the bag?______ b. Repeat the experiment in “a” but this time do it 12 times. Draw # Color
Based on your experiment, what portion of the 12 draws were GREEN? ________ Based on your experiment of 12 draws, how many green tiles do you think are in the bag?_____ c. Repeat the experiment in “a” but this time do it 18 times. Draw # Color
Based on your experiment, what portion of the 18 draws were GREEN? ________ Based on your experiment of 18 draws, how many green tiles do you think are in the bag?_____ d. Repeat the experiment in “a” but this time do it 24 times. Draw # Color Draw # Color
Based on your experiment, what portion of the 24 draws were GREEN? ________ Based on your experiment of 24 draws, how many green tiles do you think are in the bag?_____ e. Repeat the experiment in “a” but this time do it 30 times. Draw # Color Draw # Color
Based on your experiment, what portion of the 30 draws were GREEN? ________ Based on your experiment of 30 draws, how many green tiles do you think are in the bag?_____ SDUHSD Math A College Prep Module #1 – STUDENT EDITION 2017-2018
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Probability has standard notation. In this situation, we understand P(G) to mean the probability of drawing a green tile. To write P(G) we need to know the “observed frequency” and the “total number of trials.” Experimental Probability is the ratio of the observed frequency to the total number of trials:
P(G) =
observed frequency total number of trials
For each of your trials on #2, write your group’s P(G) as a simplified fraction and decimal. 2a) P(G) =
__________
__________
2d) P(G) =
__________
__________
2b) P(G) =
__________
__________
2e) P(G) =
__________
__________
2c) P(G) =
__________
__________
3. What do you think would be in the bag if: a. P(G) = 0
b.
P(G) = 1
4. Explain why the probability of GREEN tiles is a fraction (between 0 and 1).
5. How does knowing the probability of GREEN tiles help you know the probability of BLUE tiles?
6. Record a prediction of how many of the 12 tiles in your bag are GREEN.
7. NOW you can look in your bag! Count how many blue and green tiles are actually in your bag. Based on this information, what is the “theoretical” probability of drawing a green tile from your bag? Express your answer as a simplified fraction and as a decimal.
8. How did your group’s “experimental” P(G) compare with the “theoretical” P(G)?
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As groups discuss: 9. Suppose you had a bag of 1000 blue and green tiles. How many times do you think you would need to draw tiles to make an accurate prediction of the number of blue and green tiles that are actually in the bag? Explain.
Theoretical Probability:
Experimental Probability:
10. As a whole class, complete a line plot showing each group’s theoretical probability. (your group’s answer to #7)
11. Which group is most likely to have an outcome of drawing a green out of the bag? Justify your answer.
12. Which group is least likely to have an outcome of drawing a green out of the bag? Justify your answer.
13. You’re a teacher in a 7th grade math class and you want to create an experiment for your class with red, yellow and purple marbles in a bag. You want the theoretical probability of drawing a red marble to be
1 1 , the theoretical probability of drawing a yellow to be , and the theoretical probability of drawing a 4 4 1 purple to be . If you want a total of 120 marbles in the bag: 2 a. How many red marbles should you put in the bag? __________ b. How many yellow marbles should you put in the bag? __________ c. How many purple marbles should you put in the bag? __________ SDUHSD Math A College Prep Module #1 – STUDENT EDITION 2017-2018
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Finding the likelihood of an event:
14. Probability is a measure between ____ and ___ as shown on the number line, and can be written as a __________________, a _______________, or a _________________. Tell whether each event is impossible, unlikely, as likely as not, likely, or certain. Then, tell whether the probability is 0, close to 0, ½, close to 1, or 1. 15. You roll a six-sided number cube (standard die) and the number is 1 or greater.
16. You role an odd number on a standard die.
17. A bag contains 10 tiles in it and 2 of them are green. What is the probability of picking a green tile?
Find the probability of the event described below. Use probability notation and write your answer as a reduced fraction. 18. What is the probability of rolling an even number on a standard number cube?
19. What is the probability of picking a purple marble from a jar with 10 green and 8 purple marbles?
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20. Martin has a bag of marbles. He removed one marble at random, recorded the color and then placed it back in the bag. He repeated this process several times and recorded his results in the table. Find the experimental probability of drawing each color as a fraction in simplest form. Color Color Red Blue Green Yellow
Frequency 12 10 15 13
Experimental Probability
Red Blue Green Yellow
21. A spinner has three sections: red, yellow, and blue. The table shows the results of Nolan’s spins. Find the experimental probability of landing on each color. Write your answers as a fraction in simplest form. Color Color Red Yellow Blue
Frequency 10 14 6
Experimental Probability
Red Yellow Blue
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1.1B Lesson: Probability – Race to the Top* Name:
Period
1. Predict which horse you think will win (2 through 12).
2. In your group, take turns rolling two dice and give each horse their point on the histogram as they earn it. The race is over after 30 rolls of the dice.
2
3
4
5
6
7
8
9
10
11
12
3. Which horse won (had the most rolls) in your group?______ 4. List your group’s experimental probability in simplest form for each outcome: a. P(2) ________
e. P(6) ________
i. P(10) ________
b. P(3) ________
f.
P(7) ________
b. P(11) ________
a. P(4) ________
g. P(8) ________
k. P(12) ________
d. P(5) ________
h. P(9) ________
5. List the horse that won for each of the groups in your class. Which horse won the most often?
6. With your class create a class histogram that combines all the winning horses from each group. What do you notice about the histogram?
2 3 4 5 6 7 8 7. Which horse won the most often for all the groups? Why?
9
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11
12
18
8. Do you think that this game is fair? Why or why not?
9. What are all the possible outcomes when you roll two dice? In your group, organize these possible outcomes on the given chart.
2
3
4
5
6
7
8
9
10
11
12
10. How many total outcomes did you get? Explain the system you used to get all those outcomes.
11. Use the above information to determine the probability in simplest form for each outcome: P(1)_______
P(7)________
P(2)_______
P(8)________
P(3)_______
P(9)________
P(4)_______
P(10) _______
P(5)_______
P(11)_______
P(6)_______
P(12)_______
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1.1C Lesson: Sample Spaces, Outcomes, and Probabilities* Name:
Period
Tree Diagram: Sample Space:
For #1-3, use a tree diagram to answer the following questions. Write all fractional answers in simplest form. 1. In your closet you choose from a red shirt and a blue shirt and you choose from jeans, shorts, and sweatpants how many different outfit combinations can you make?
a. How many outcomes are in this sample space? __________ b. List the outcomes: c. What is the P(shorts)? __________ d. What is the P(blue shirt)? __________ 2. You flip a coin three times. How many combinations are possible?
a. How many outcomes are in this sample space? __________ b. What is the P(three tails)? __________ c. What is the P(exactly two tails)? __________ SDUHSD Math A College Prep Module #1 – STUDENT EDITION 2017-2018
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3. A deli prepares sandwiches with one type of bread (white or wheat), one type of meat (ham, turkey, or chicken), and one type of cheese (cheddar or Swiss). How many combinations are possible?
a. How many outcomes are in this sample space? __________ b. What is the P(ham)? __________ c. What is the P(swiss)? __________
Fundamental Counting Principle:
For #4-7, use the Fundamental Counting Principle to answer the following questions. Write all fractional answers in simplest form. 4. You are trying to decide what to wear to the school dance. You know you want to wear jeans, but are unsure of the shirt and shoes. If you are deciding between 16 shirts and 3 pairs of shoes and if you try on all combinations for your friend to help you decide, how many would she see? Show work that justifies your answer.
5. Grace loves to eat salad. How many salads can she put together if she can pick out one type of lettuce from 2 choices, one vegetable from 4 choices and one dressing from 7 choices? Show work that justifies your answer.
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6. California used to issue license plates consisting of two letters followed by three numbers. There are 26 letters and 10 digits and the letters and digits may be repeated. How many possible license plates could be issued with two letters followed by three numbers? Show work that justifies your answer.
7. An apartment complex offers apartments with four different options. If one is selected from each option, how many combinations do I have to choose from? Show work that justifies your answer. A. One bedroom Two bedrooms Three bedrooms B. One bathroom Two bathrooms C. First floor Second floor D. Lake view Golf course view No special view
a. How many outcomes are in this sample space? __________ b. What is the P(first floor)? __________ c. What is the P(lake view)? __________ d. What is the P(first floor and lake view)? __________ e. What is the P(first floor or lake view)? __________
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Section 1.2: Understand/apply equivalence in rational number forms. Convert between forms (fraction, decimal, percent).* Section Overview: In this section students solidify and practice rational number sense through the careful review of fractions, decimals and percent. The two key objectives of this section are a) students should be confidently able to articulate with words, models and symbols the relationship among equivalent fractions, decimals, and percent and b) students should understand and use models to find portions of different wholes. The concept of equivalent fractions naturally leads students to the issues of ordering and estimation. Ordering positive and negative fractions will be connected to the number line. It is important that students develop estimation skills in conjunction with both ordering and operating on positive and negative rational numbers. Lastly, students look at percent as being a fraction with a denominator of 100. Percent and fraction contexts in this section should be approached intuitively with models. In 1.3 students will begin to transition to writing numeric expressions. Concepts and Skills to be Mastered (from standards) 1. 2. 3. 4. 5. 6. 7. 8.
Express probability using appropriate fractions, ratios, decimals, and percents. Find the percent of a quantity using a model. Express and convert between rational numbers in different forms. Express fractions, decimals and percents as related to two-dimensional area or area models. Draw models to show equivalence among fractions and rational numbers. Solve problems with rational numbers using models. Solve problems with rational numbers using estimation. Compare rational numbers in different forms.
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1.2A Lesson: Bar Models with Fractions and Decimals* Name:
Period:
Draw a bar model to represent each of the following fractions: ! ! 1. 2. !"
3.
!
!"
4.
!"
5.
! !"
What do you notice about the bar models for #1 and #3 and for #2 and #4? What is similar? What is different?
Reduce/simplify each fraction. Draw a bar model to show the equivalence between the original fraction and the reduced one: 6.
!
7.
!"
!" !"
8.
! !"
Find an equivalent fraction for each. Draw a bar model to show the equivalence between the original fraction and the new one. 9.
1 ? = 3 9
10.
3 6 = 7 ?
SDUHSD Math A College Prep Module #1 – STUDENT EDITION 2017-2018
11.
4 ? = 5 25
24
Change each to a mixed number. Draw a bar model to show the equivalence between the original fraction and the new one. 12.
13.
14.
! !
!" !
!" !
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1.2B Lesson: Rational Number Ordering and Estimation* Name:
Period:
1. Plot each fraction on the number line below. a.
1 1 3 2 , , , 2 3 4 5
b. Compare the fractions using <, >, or =.
1 1 ____ 2 3
2 3 _____ 5 4
2. What differences do you observe when comparing the fractions in part b?
3. How does the number line help you determine which number is larger?
4. Classify these fractions as close to 0, close to
1 , or close to 1. 2
1 1 5 5 7 1 5 3 1 , , , , , , , , 2 9 8 6 8 5 9 8 7
Close to 0
Close to
1 2
Close to 1
5. Order the fractions from least to greatest. Fractions: ______, ______, ______, ______, ______, ______, ______, ______, ______
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6. Approximate where each value is located on the number line below. State a “common” fraction and decimal you can use to help you find the approximate location for each. The first one is done for you.
a. 0.32 ≈0.3 or ≈ d. 0.98
! !
b. 0.67
c. 0.76
e. 0.18
f. 0.06
Plot each fraction on the number line. Fill in the blank with a <, >, or =. 7. 4
2 3 _____ 4 7 10
Plot each fraction on the number line. Fill in the blank with <, >, or =. How do you know your answer is correct? Justify your answer. 8. 0.14 _____0.14 Justification: >
9. 0.15_____
3 20
Justification:
10. 2
2 _____2.6 3
Justification:
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11. Is 0.74 to the left or right of
3 on a number line? Explain. 4
12. Is 1.26 to the left or right of 1
1 on a number line? Explain. 4
13. For each problem below, circle which number is bigger. Explain how you know. a. 0.4 and 0.04
b. 0.27 and 0.3
c. 0.1 and 0.09
14. Without using a calculator, determine which fraction is bigger in each pair. Justify your answer with a picture and words. a.
1 3
or
1 2
b.
3 7
or
3 5
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1.2C Lesson: Probability, Fractions, Percentage, & Ratio* Name:
Period:
1. A bag contains 100 marbles. The table below shows how many red, blue, green and yellow marbles are in the bag. Use that data to complete the table below. The first row is completed for you.
Color of Marble
a.
Red
b.
Blue
c.
Green
d.
Yellow
e.
Orange
f.
Red or blue
g.
Red, blue, green, or yellow
Color
Number of Marbles
Red Blue Green Yellow
16 24 45 15
Probability of drawing the colored marble
Percentage of colored marble within all marbles
16 4 = 100 25
16%
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Use a model to solve the following questions. 2. Find 60% of 180.
3. Find 40% of 80.
4. Find 25% of 324.
5. What percent of 80 is 60?
6. What percent of 120 is 48?
7. What percent of 95 is 19?
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8. 30 is 25 percent of what number?
9. 12 is 10 percent of what number?
10. 45 is 15 percent of what number?
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1.2D Lesson: Rational Numbers in Applications with Models* Name:
Period:
Percent and fraction questions: Use a model to find the solutions. 1. 60 is 40% of what number?
2. There are 36 students in a math class.
3 of the students take an art class after their math class and the 4
rest take a social studies class. How many students take art after math?
3. You get 80% correct on a history quiz with 150 questions. How many questions did you get correct?
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4. Juan earned money for creating a webpage for a local business. He used new shoes and
1 of the money he earned for 2
2 of the rest for music. He has $20 left. How much money did he earn for his work? 3
5. Lydia volunteers with an organization that helps older citizens take care of their yards. 75% of the volunteers in the organization are 20-30 years old. Of the remaining portion, 75% are over 30 and 25% are under 20. If there are 15 people under 20, how many people are in the organization?
6. There are 360 7th grade students at Eisenhower Middle School. One-fourth of the students went to Clermont Elementary. Of the rest, half went to Central Elementary and the others came from a variety of other elementary schools. How many students came from Central Elementary?
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7. A snowboard at a local shop normally costs $450. Over Labor Day weekend, the snowboard is on sale for 50% off. Customers who make their purchase before 8:00 AM earn an additional 10% off of the sale price. If Mia buys the snowboard before 8:00 AM, how much will she pay?
1 1 of their money on payroll, of the 3 2 1 remaining amount went back into the business to purchase inventory and pay for the facility, and of 3
8. A local business is taking out a loan. They found that they spent
the money that was left after that went to paying off their original loan. If they have $100,000 left of the loan, what was the original amount of the loan?
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Section 1.3: Solve Percent Problems Including Discounts, Interest, Taxes, Tips, and Percent Increase or Decrease.* Section Overview: In this section, students continue to solve contextual problems with fractions, decimals and percent but begin to transition from relying solely on models to writing numeric expressions. In future modules students will extend their understanding by writing equations and proportions using variables. Concepts and Skills to be Mastered (from standards) 1. 2. 3. 4.
Use models to solve problems involving percent and fractions. Solve percent problems involving discounts, interest, taxes, tips, etc. Solve percent problems involving percent increase and decrease. Develop algebraic expressions and equations from percent and fraction models.
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1.3A Lesson Model Percent and Fraction Problems* Name:
Period:
Mark Up:
Discount:
Use a model to solve each of the following multi-step problems. Then write a number sentence that reflects your model and answer. 1. What is 60% of 80? a.
Solve by using a model.
b. Solve by writing and using a number sentence.
2. 72 is 60% of what number? a. Solve by using a model.
b. Solve by writing and using a number sentence.
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3. A refrigerator costs $1200 wholesale. If the mark-up on the refrigerator is 20%, what is the new price? a. Solve by using a model.
b. Solve by writing and using a number sentence.
4. Rico's resting heart rate is 50 beats per minute. His target exercise rate is 150% of his resting rate. What is his target rate? a. Solve by using a model.
b. Solve by writing and using a number sentence.
5. A pair of boots was originally priced at $200. The store put them on sale for 25% off. A month later, the boots were reduced an additional 50% off the previous sale price. What is the price now? $75 a.
Solve by using a model.
b. Solve by writing and using a number sentence.
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6. It used to take Nyah 10 minutes to walk to school. Now it takes her 8 minutes. What is her percent decrease? a.
Solve by using a model.
b.
Solve by writing and using a number sentence.
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1.3B Lesson: Percent and Fraction Problems* Name:
Period:
Write a number sentence to solve the following problems. You may also use a model if helpful. 1. What percent of 40 is 25?
2. What is 150% of 90?
3. 45 is 1.5% of what number?
4. Nineteen members, or 38%, of the surf club are going on a surf trip. Find the total number of member in the club.
5. Store AA is selling a pair of shoes for 20% off the store’s price of $25. The same shoes are going for 40% off Store BB’s regular price of $30. Which pair of shoes is the better buy? Explain your choice.
6. Marcie went out for dinner with her friend. The dinner cost $25. Tax is 5% and Marcie wants to leave a 15% tip on the pre-tax amount. How much will Marcie pay all together for dinner?
7. 100 is increased by 10%. The result is decreased by 10%. Is the final result 100? Explain your reasoning.
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8. A jacket that originally cost $100 was discounted 20% for a 4th of July sale. After the sale was over, the jacket was marked-up 20%. How much does the jacket cost after the sale was over?
9. A music store’s percent of markup is 67%. A DVD costs the store $10.15. Find the markup. Then find how much the store will sell the DVD to its customers.
10. A pair of running shoes was originally $75. They are on sale for $60. What is the percent of discount?
11. A small business’s profits in 2011 were $120,000. In 2012 they decreased 25%. How much did the business make in profits in 2012? 12. A surfboard is marked $700 at Hansen’s. The store is offering 20% off for their Fourth of July sale. They are offering an additional 20% off if you come early to shop between the hours of 8AM-9AM. How does this deal compare to 40% off the surfboard that they will offer for their Labor Day sale? When should I buy the surfboard?
13. How can you explain why the deals in #12 above are not the same?
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